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<h1 id="CR-Nonconforming-Element-for-Poisson-Equation-in-3D">CR Nonconforming Element for Poisson Equation in 3D<a class="anchor-link" href="#CR-Nonconforming-Element-for-Poisson-Equation-in-3D">&#182;</a></h1>
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<p>This example is to show the rate of convergence of the CR Nonconforming finite element approximation of the Poisson equation on the unit cube:</p>
$$- \Delta u = f \; \hbox{in } (0,1)^3$$<p>for the following boundary conditions</p>
<ul>
<li>Non-empty Dirichlet boundary condition: $u=g_D \hbox{ on }\Gamma_D, \nabla u\cdot n=g_N \hbox{ on }\Gamma_N.$</li>
<li>Pure Neumann boundary condition: $\nabla u\cdot n=g_N \hbox{ on } \partial \Omega$.</li>
<li>Robin boundary condition: $g_R u + \nabla u\cdot n=g_N \hbox{ on }\partial \Omega$.</li>
</ul>
<p><strong>References</strong>:</p>
<ul>
<li><a href="femdoc.html">Quick Introduction to Finite Element Methods</a></li>
<li><a href="http://www.math.uci.edu/~chenlong/226/Ch2FEM.pdf">Introduction to Finite Element Methods</a></li>
<li><a href="http://www.math.uci.edu/~chenlong/226/Ch3FEMCode.pdf">Progamming of Finite Element Methods</a></li>
</ul>
<p><strong>Subroutines</strong>:</p>

<pre><code>- Poisson3CR
- cubePoisson
- femPoisson3
- Poisson3CRfemrate

</code></pre>
<p>The method is implemented in <code>Poisson3CR</code> subroutine and tested in <code>cubePoissonCR</code>. Together with other elements (P1,P2,Q1,WG,CR), <code>femPoisson3</code> provides a concise interface to solve Poisson equation. The CR element is tested in <code>Poisson3CRfemrate</code>. This doc is based on <code>Poisson3CRfemrate</code>.</p>

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<h2 id="CR-Nonconforming-Element">CR Nonconforming Element<a class="anchor-link" href="#CR-Nonconforming-Element">&#182;</a></h2><p>We explain degree of freedoms and basis functions for Crouzeix-Raviart nonconforming P1 element on a tetrahedron. The dofs are associated to faces. Given a mesh, the required data structure can be constructured by</p>

<pre><code>[elem2face,face] = dof3face(elem);

</code></pre>
<h3 id="Local-indexing">Local indexing<a class="anchor-link" href="#Local-indexing">&#182;</a></h3>
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<div class=" highlight hl-matlab"><pre><span></span><span class="n">node</span> <span class="p">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">;</span> <span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">;</span> <span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">];</span>
<span class="n">elem</span> <span class="p">=</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">4</span><span class="p">];</span>
<span class="n">face</span> <span class="p">=</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">3</span> <span class="mi">4</span><span class="p">;</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">4</span><span class="p">;</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">4</span><span class="p">;</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">];</span>
<span class="n">showmesh3</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span> <span class="n">view</span><span class="p">([</span><span class="o">-</span><span class="mi">26</span> <span class="mi">10</span><span class="p">]);</span>
<span class="n">findnode3</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
<span class="n">findelem</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">face</span><span class="p">);</span>
</pre></div>

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<h3 id="A-Local-Basis">A Local Basis<a class="anchor-link" href="#A-Local-Basis">&#182;</a></h3><p>The 4 Lagrange-type bases functions are denoted by $\phi_i, i=1:4$, i.e. $\phi_i(m_j)=\delta _{ij},i,j=1:4$, where $m_i$ is the center of the i-th face. In barycentric coordinates, they are:</p>
$$\phi_i = 1- 2\lambda_i,\quad \nabla \phi_i = -2\nabla \lambda_i,\quad i =1:4.$$<p>When transfer to the reference triangle formed by $(0,0,0),(1,0,0),(0,1,0),(0,0,1)$, the local bases in x-y-z coordinate can be obtained by substituting</p>
$$\lambda _1 = x, \quad \lambda _2 = y, \quad \lambda _3 = z, \quad \lambda_4 = 1-x-y-z.$$<p></p>

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<h2 id="Mixed-boundary-condition">Mixed boundary condition<a class="anchor-link" href="#Mixed-boundary-condition">&#182;</a></h2>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">%% Setting</span>
<span class="p">[</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">]</span> <span class="p">=</span> <span class="n">cubemesh</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span><span class="mf">0.5</span><span class="p">);</span> 
<span class="n">mesh</span> <span class="p">=</span> <span class="n">struct</span><span class="p">(</span><span class="s">&#39;node&#39;</span><span class="p">,</span><span class="n">node</span><span class="p">,</span><span class="s">&#39;elem&#39;</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">option</span><span class="p">.</span><span class="n">L0</span> <span class="p">=</span> <span class="mi">1</span><span class="p">;</span>
<span class="n">option</span><span class="p">.</span><span class="n">maxIt</span> <span class="p">=</span> <span class="mi">4</span><span class="p">;</span>
<span class="n">option</span><span class="p">.</span><span class="n">elemType</span> <span class="p">=</span> <span class="s">&#39;CR&#39;</span><span class="p">;</span>
<span class="n">option</span><span class="p">.</span><span class="n">printlevel</span> <span class="p">=</span> <span class="mi">1</span><span class="p">;</span>
<span class="n">option</span><span class="p">.</span><span class="n">plotflag</span> <span class="p">=</span> <span class="mi">1</span><span class="p">;</span>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">%% Non-empty Dirichlet boundary condition.</span>
<span class="n">pde</span> <span class="p">=</span> <span class="n">sincosdata3</span><span class="p">;</span>
<span class="n">mesh</span><span class="p">.</span><span class="n">bdFlag</span> <span class="p">=</span> <span class="n">setboundary3</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="s">&#39;Dirichlet&#39;</span><span class="p">,</span><span class="s">&#39;~(x==0)&#39;</span><span class="p">,</span><span class="s">&#39;Neumann&#39;</span><span class="p">,</span><span class="s">&#39;x==0&#39;</span><span class="p">);</span>
<span class="n">femPoisson3</span><span class="p">(</span><span class="n">mesh</span><span class="p">,</span><span class="n">pde</span><span class="p">,</span><span class="n">option</span><span class="p">);</span>
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<pre>Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     6528,  #nnz:    23040, smoothing: (1,1), iter: 17,   err = 7.80e-09,   time =  0.1 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    50688,  #nnz:   190464, smoothing: (1,1), iter: 17,   err = 9.10e-09,   time = 0.25 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:   399360,  #nnz:  1548288, smoothing: (1,1), iter: 17,   err = 9.78e-09,   time =  2.6 s
Table: Error
 #Dof        h        ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}

   864   2.500e-01   1.57877e-02   5.75062e-01   1.15791e-01   1.49758e-02
  6528   1.250e-01   4.18680e-03   2.93247e-01   5.28394e-02   4.03690e-03
 50688   6.250e-02   1.06111e-03   1.47388e-01   2.58754e-02   1.05299e-03
399360   3.125e-02   2.66154e-04   7.37911e-02   1.28726e-02   2.66527e-04

Table: CPU time
 #Dof    Assemble     Solve      Error      Mesh    

   864   1.10e-01   7.92e-03   1.00e-01   2.00e-02
  6528   5.00e-02   1.02e-01   5.00e-02   1.00e-02
 50688   2.10e-01   2.47e-01   2.10e-01   1.00e-01
399360   2.42e+00   2.60e+00   1.47e+00   0.00e+00

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<h2 id="Pure-Neumann-boundary-condition">Pure Neumann boundary condition<a class="anchor-link" href="#Pure-Neumann-boundary-condition">&#182;</a></h2><p>When pure Neumann boundary condition is posed, i.e., $-\Delta u =f$ in $\Omega$ and $\nabla u\cdot n=g_N$ on $\partial \Omega$, the data should be consisitent in the sense that $\int_{\Omega} f \, dx + \int_{\partial \Omega} g \, ds = 0$. The solution is unique up to a constant. A post-process is applied such that the constraint $\int_{\Omega}u_h dx = 0$ is imposed.</p>

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<div class=" highlight hl-matlab"><pre><span></span><span class="c">%% Pure Neumann boundary condition.</span>
<span class="n">option</span><span class="p">.</span><span class="n">plotflag</span> <span class="p">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="n">mesh</span><span class="p">.</span><span class="n">bdFlag</span> <span class="p">=</span> <span class="n">setboundary3</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="s">&#39;Neumann&#39;</span><span class="p">);</span>
<span class="n">femPoisson3</span><span class="p">(</span><span class="n">mesh</span><span class="p">,</span><span class="n">pde</span><span class="p">,</span><span class="n">option</span><span class="p">);</span>
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<pre>Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     6528,  #nnz:    24957, smoothing: (1,1), iter: 20,   err = 7.43e-09,   time = 0.092 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    50688,  #nnz:   198141, smoothing: (1,1), iter: 21,   err = 4.01e-09,   time = 0.24 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:   399360,  #nnz:  1579005, smoothing: (1,1), iter: 22,   err = 4.23e-09,   time =  2.7 s
Table: Error
 #Dof        h        ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}

   864   2.500e-01   2.24448e-02   5.86299e-01   1.53064e-01   6.19341e-02
  6528   1.250e-01   5.72292e-03   2.94670e-01   5.84934e-02   1.63182e-02
 50688   6.250e-02   1.43932e-03   1.47567e-01   2.66183e-02   4.13880e-03
399360   3.125e-02   3.60390e-04   7.38135e-02   1.29661e-02   1.03857e-03

Table: CPU time
 #Dof    Assemble     Solve      Error      Mesh    

   864   4.00e-02   1.86e-03   2.00e-02   1.00e-02
  6528   3.00e-02   9.16e-02   3.00e-02   1.00e-02
 50688   2.10e-01   2.37e-01   1.30e-01   7.00e-02
399360   2.74e+00   2.70e+00   1.41e+00   0.00e+00

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<h2 id="Robin-boundary-condition">Robin boundary condition<a class="anchor-link" href="#Robin-boundary-condition">&#182;</a></h2>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">%% Pure Robin boundary condition.</span>
<span class="n">pde</span> <span class="p">=</span> <span class="n">sincosRobindata3</span><span class="p">;</span>
<span class="n">mesh</span><span class="p">.</span><span class="n">bdFlag</span> <span class="p">=</span> <span class="n">setboundary3</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="s">&#39;Robin&#39;</span><span class="p">);</span>
<span class="n">femPoisson3</span><span class="p">(</span><span class="n">mesh</span><span class="p">,</span><span class="n">pde</span><span class="p">,</span><span class="n">option</span><span class="p">);</span>
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<pre>Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     6528,  #nnz:    24960, smoothing: (1,1), iter: 17,   err = 3.83e-09,   time = 0.074 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    50688,  #nnz:   198144, smoothing: (1,1), iter: 17,   err = 7.40e-09,   time = 0.31 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:   399360,  #nnz:  1579008, smoothing: (1,1), iter: 17,   err = 7.34e-09,   time =    3 s
Table: Error
 #Dof        h        ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}

   864   2.500e-01   1.70968e-02   5.78078e-01   1.24451e-01   2.87371e-02
  6528   1.250e-01   4.51894e-03   2.93659e-01   5.40040e-02   7.24170e-03
 50688   6.250e-02   1.14572e-03   1.47441e-01   2.60230e-02   1.82017e-03
399360   3.125e-02   2.87439e-04   7.37977e-02   1.28910e-02   4.57164e-04

Table: CPU time
 #Dof    Assemble     Solve      Error      Mesh    

   864   5.00e-02   9.15e-04   0.00e+00   0.00e+00
  6528   3.00e-02   7.44e-02   2.00e-02   1.00e-02
 50688   2.00e-01   3.09e-01   1.70e-01   7.00e-02
399360   2.16e+00   3.05e+00   1.48e+00   0.00e+00

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<h2 id="Conclusion">Conclusion<a class="anchor-link" href="#Conclusion">&#182;</a></h2><p>The optimal rate of convergence of the H1-norm (1st order) and L2-norm (2nd order) is observed. No superconvergence for $\|\nabla u_I - \nabla u_h\|$.</p>
<p>MGCG converges uniformly in all cases.</p>

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